This paper addresses the important goal of quantifying the threat of linking external records to public Online Social Networks (OSN) user profiles, by providing a method to estimate the uniqueness of such profiles and by studying the amount of information carried by public profile attributes. Our first contribution is to leverage the Ads audience estimation platform of a major OSN to compute the information surprisal (IS) based uniqueness of public profiles, independently from the used profiles dataset. Then, we measure the quantity of information carried by the revealed attributes and evaluate the impact of the public release of selected combinations of these attributes on the potential to identify user profiles. Our measurement results, based on an unbiased sample of more than 400 thousand Facebook public profiles, show that, when disclosed in such profiles, current city has the highest individual attribute potential for unique identification and the combination of gender, current city and age can identify close to 55% of users to within a group of 20 and uniquely identify around 18% of users. We envisage the use of our methodology to assist both OSNs in designing better anonymization strategies when releasing user records and users to evaluate the potential for external parties to uniquely identify their public profiles and hence make it easier to link them with other data sources.